TY - JOUR
T1 - Non-residues and primitive roots in beatty sequences
AU - Banks, William D.
AU - Shparlinski, Igor E.
PY - 2006/6
Y1 - 2006/6
N2 - We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1, 2, 3, . . .}, where α, β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p1/2+ε ≤ N ≤ p, then among the first N elements of Bα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.
AB - We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1, 2, 3, . . .}, where α, β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p1/2+ε ≤ N ≤ p, then among the first N elements of Bα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.
UR - http://www.scopus.com/inward/record.url?scp=33746086327&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:33746086327
VL - 73
SP - 433
EP - 443
JO - Bulletin of the Australian Mathematical Society
JF - Bulletin of the Australian Mathematical Society
SN - 0004-9727
IS - 3
ER -